Best Known (241−36, 241, s)-Nets in Base 2
(241−36, 241, 624)-Net over F2 — Constructive and digital
Digital (205, 241, 624)-net over F2, using
- 21 times duplication [i] based on digital (204, 240, 624)-net over F2, using
- t-expansion [i] based on digital (203, 240, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 40, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 40, 104)-net over F64, using
- t-expansion [i] based on digital (203, 240, 624)-net over F2, using
(241−36, 241, 2171)-Net over F2 — Digital
Digital (205, 241, 2171)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 2171, F2, 3, 36) (dual of [(2171, 3), 6272, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2241, 2741, F2, 3, 36) (dual of [(2741, 3), 7982, 37]-NRT-code), using
- strength reduction [i] based on linear OOA(2241, 2741, F2, 3, 37) (dual of [(2741, 3), 7982, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2241, 8223, F2, 37) (dual of [8223, 7982, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 8225, F2, 37) (dual of [8225, 7984, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(2235, 8193, F2, 37) (dual of [8193, 7958, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(2209, 8193, F2, 33) (dual of [8193, 7984, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2241, 8225, F2, 37) (dual of [8225, 7984, 38]-code), using
- OOA 3-folding [i] based on linear OA(2241, 8223, F2, 37) (dual of [8223, 7982, 38]-code), using
- strength reduction [i] based on linear OOA(2241, 2741, F2, 3, 37) (dual of [(2741, 3), 7982, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2241, 2741, F2, 3, 36) (dual of [(2741, 3), 7982, 37]-NRT-code), using
(241−36, 241, 80992)-Net in Base 2 — Upper bound on s
There is no (205, 241, 80993)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3 534033 446821 796773 274364 298810 821683 552818 746354 417378 757410 849056 149624 > 2241 [i]